Mixtures ideal

Consider an ideal gas consisting of A i particles of substance 1, N2 particles of substance 2. and Nk particles of substance K. The number K of different substances can be arbitrarily high. 

With a quadratic term in the Hamiltonian for each momentum term, the integral in Eq. 10.2 can be determined to give 

The thermodynamic properties of the ideal mixture (Fig. 10.1) can now be determined. The Helmholtz free energy of the mixture is 

Using the particle density of each substance, Pk = Nk/V, and rearranging yields 

Finally, the chemical potential of each substance in an ideal gas mixture is defined as 

Depending on the temperature, pressure, and identity of the constituents of a liquid mixture, Raoult s law for fugacity may hold for constituent i at all liquid compositions, or over only a limited composition range when xt is close to unity. 

An ideal liquid mixture is defined as a liquid mixture in which, at a given temperature and pressure, each constituent obeys Raoult s law for fugacity (Eq. 9.4.3 or 9.4.5) over the entire range of composition. Equation 9.4.3 applies only to a volatile constituent, whereas Eq. 9.4.5 applies regardless of whether the constituent is volatile. 

Few liquid mixtures are found to approximate the behavior of an ideal liquid mixture. In order to do so, the constituents must have similar molecular size and structure, and the pure liquids must be miscible in all proportions. Benzene and toluene, for instance, satisfy these requirements, and liquid mixtures of benzene and toluene are found to obey Raoult s law quite closely. In contrast, water and methanol, although miscible in all proportions, form liquid mixtures that deviate considerably from Raoult s law. The most conunonly encountered situation for mixtures of organic liquids is that each constituent deviates from Raoult s law behavior by having a higher fugacity than predicted by Eq. 9.4.3—a positive deviation from Raoult s law. 

Similar statements apply to ideal solid mixtures. In addition, a relation with the same form as Eq. 9.4.5 describes the chemical potential of each constituent of an ideal gas mixture, as the following derivation shows. In an ideal gas mixture at a given T and p, the chemical potential of substance i is given by Eq. 9.3.5  

Here yi is the mole fraction of i in the gas. The chemical potential of the pure ideal gas 

---

In principle, extractive distillation is more useful than azeotropic distillation because the process does not depend on the accident of azeotrope formation, and thus a greater choice of mass-separating agent is, in principle, possible. In general, the solvent should have a chemical structure similar to that of the less volatile of the two components. It will then tend to form a near-ideal mixture with the less volatile component and a nonideal mixture with the more volatile component. This has the effect of increasing the volatility of the more volatile component. 

Figure A2.5.3. Typical liquid-gas phase diagram (temperature T versus mole fraction v at constant pressure) for a two-component system in which both the liquid and the gas are ideal mixtures. Note the extent of the two-phase liquid-gas region. The dashed vertical line is the direction x = 1/2) along which the fiinctions in figure A2.5.5 are detemiined.

Figure A2.5.22. The experimental heat eapaeity of a p-brass (CiiZn) alloy eontaining 48.9 atomie pereent Zn as measured by Moser (1934). The dashed line is ealeulated from the speeifie heats of Cu and Zn assuming an ideal mixture. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 4, figure 4. Copyright (1938) by the Arneriean Physieal Soeiety.

However, in the study of thermodynamics and transport phenomena, the behavior of ideal gases and gas mixtures has historically provided a norm against which their more unruly brethren could be measured, and a signpost to the systematic treatment of departures from ideality. In view of the complexity of transport phenomena in multicomponent mixtures a thorough understanding of the behavior of ideal mixtures is certainly a prerequisite for any progress in understanding non-ideal systems. 

This result should be compared with Eq. (8.28) for the case of the ideal mixture. It is reassuring to note that for n = 1, Eq. (8.36) reduces to Eq. (8.28). Next let us consider whether a change of notation will clarify Eq. (8.36) still more. Recognizing that the solvent, the repeat unit, and the lattice site all have the same volume, we see that Ni/N is the volume fraction occupied by the solvent in the mixture and nN2/N is the volume fraction of the polymer. Letting be the volume fraction of component i, we see that Eq. (8.36) becomes... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

The three-phase region of D2—DT—T2 has been studied (12). Relative volatilities for the isotopic system deuterium—deuterium tritide—tritium have been found (13) to be 5—6% below the values predicted for ideal mixtures. 

Irreversible processes are mainly appHed for the separation of heavy stable isotopes, where the separation factors of the more reversible methods, eg, distillation, absorption, or chemical exchange, are so low that the diffusion separation methods become economically more attractive. Although appHcation of these processes is presented in terms of isotope separation, the results are equally vaUd for the description of separation processes for any ideal mixture of very similar constituents such as close-cut petroleum fractions, members of a homologous series of organic compounds, isomeric chemical compounds, or biological materials. 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

The efficiency of a distillation apparatus used for purification of liquids depends on the difference in boiling points of the pure material and its impurities. For example, if two components of an ideal mixture have vapour pressures in the ratio 2 1, it would be necessary to have a still with an efficiency of at least seven plates (giving an enrichment of 2 = 128) if the concentration of the higher-boiling component in the distillate was to be reduced to less than 1% of its initial value. For a vapour pressure ratio of 5 1, three plates would achieve as much separation. 

In an ideal mixture the partial pressure p is proportional to the mole fraction of the component in the gas phase ... 

For this expression to be valid, in cell A components 1 and 2 must be identical in all respects, so it is a rather special case of an ideal mixture. They are however, allowed to interact differently with the membrane, as described above, xa is the mole fraction of the solute in cell A, while p and p are the number densities of cells A and B respectively. The method was extensively tested against both Monte Carlo and equations of state for LJ particles, and the values of the chemical potential were found to be satisfactory. The method can also be extended to mixtures [29] by making... 

There are many mixtures of liquids that do not follow Raoult s Law, which represents the performance of ideal mixtures. For those systems following the ideal gas law and Raoult s Law for the liquid, for each component. 

For non-ideal mixtures the minimum L/V may be as indicated in Figure 8-15, and hence not fixed as indicated above. 

Then the volume fraction of a component is calculated assuming an ideal mixture. 

Equation (7.6) is the starting point for deriving equations for Amix2[j[, the change in Zm for forming an ideal mixture. For the ideal solution, 7r.( = 1 and equation (7.6) becomes... 

The entropy change to form an ideal mixture from the pure components is obtained by differentiating equation (7.7) with respect to T. Since x, is independent of T, the result is... 

Figure 7.1 Entropy, enthalpy, and Gibbs free energy changes at T= 298.15 K for forming one mole of an ideal mixture from the components,...

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

Effect of Pressure on Solid + Liquid Equilibrium Equation (6.84) is the starting point for deriving an equation that gives the effect of pressure on (solid + liquid) phase equilibria for an ideal mixture in equilibrium with a pure... 

Solid + Liquid Equilibria in Less Ideal Mixtures We should not be surprised to find that the near-ideal (solid + liquid) phase equilibria behavior shown in Figures 8.20 and 8.21 for (benzene + 1,4-dimethylbenzene) is unusual. Most systems show considerably larger deviations. For example, Figure 8.22 shows the phase diagram for. vin-C Hw +. The solid line is the fit of the... 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Equation 10.96 does not apply to either electrolytes or to concentrated solutions. Reid, PRAUSNITZ and Sherwood"7 discuss diffusion in electrolytes. Little information is available on diffusivides in concentrated solutions although it appears that, for ideal mixtures, the product /xD is a linear function of the molar concentration. 

The vapor of an ideal mixture of two volatile liquids is richer than the liquid in the more volatile component. The contribution of each component to the total vapor pressure and its mole fraction in the vapor can be calculated by combining Raoulfs law and Dalton s law. 

For ideal mixtures, relative volatility is the ratio of vapor pressures ai2 = P2I Pi. 

Godzik A, Kolinski A, Skolnick J. Are proteins ideal mixtures of amino acids Analysis of energy parameter sets. Protein Sci 1995 4 2107-17. 

When dealing with an ideal mixture of gases (that do not behave ideally themselves), the fugacity of gas X is given by... 

The inaccuracies of the approach lead the present writer to disagree with the author s conclusions on the UNIFAC approach, which according to him is better than the one of comparing the mixture with an ideal mixture. 

The Gibbs free energy (computed in the harmonic approximation) were converted from the 1 atm standard state into the standard state of molar concentration (ideal mixture at 1 molL-1 and 1 atm). 

---